The Chain Rule formula is a formula for computing the derivative of the composition of two or more functions. Chain rule in differentiation is defined for composite functions. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition.
d/dx [f(g(x))] = f'(g(x)) g'(x)
What is Chain Rule Formula?
The Chain Rule Formula is as follows –Â
\[\LARGE \frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}\]
Solved Examples on Chain Rule Formula
Example 1:Â Differentiate y = cos x2
Solution:
Given,
y = cos x2
Let u = x2, so that y = cos u
Therefore;
\(\begin{array}{l}\frac{du}{dx}=2x\end{array} \)
\(\begin{array}{l}\frac{dy}{du} = -sin u\end{array} \)
And so, the chain rule says:
\(\begin{array}{l}\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}\end{array} \)
\(\begin{array}{l}\frac{dy}{dx}= -sin u \times 2x\end{array} \)
= -2x sin x2
Example 2:Â
Differentiate f(x) = (1 + x2)5.
Solution:Â
Using the Chain rule,
dy/dx = dy/du â‹… du/dx
Let us take y = u5Â and u = 1 + x2
Then dy/du = d/du (u5) = 5u4
du/dx = d/dx (1 + x2Â )= 2x
dy/dx = 5u4â‹…2x = 5(1 + x2)4â‹…2x
= 10x(1 + x2)4
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